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Hilbert s Mistake Edward Nelson Department of Mathematics Princeton University 1 Abstract Hilbert was at heart a Platonist No one shall expel us from the paradise that Cantor has created for us His formalism was primarily a tactic in his battle against Brouwer s intuitionism His mistake was to pose the problem of showing that mathematics beginning with Peano Arithmetic is consistent rather than to ask whether it is consistent In this talk I give reasons for taking seriously the possibility that contemporary mathematics including Peano Arithmetic may indeed be inconsistent 2 Potential vs Completed Infinity Let us distinguish between the genetic in the dictionary sense of pertaining to origins and the formal Numerals terms containing only the unary function symbol S and the constant 0 are genetic they are formed by human activity All of mathematical activity is genetic though the subject matter is formal Numerals constitute a potential infinity Given any numeral we can construct a new numeral by prefixing it with S Now imagine this potential infinity to be completed Imagine the inexhaustible process of constructing numerals somehow to have been finished and call the result the set of all numbers denoted by N Thus N is thought to be an actual infinity or a completed infinity This is curious terminology since the etymology of infinite is not finished 3 We were warned Aristotle Infinity is always potential never actual Gauss I protest against the use of infinite magnitude as something completed which is never permissible in mathematics We ignored the warnings With the work of Dedekind Peano and Cantor above all completed infinity was accepted into mainstream mathematics Mathematics became a faith based initiative Try to imagine N as if it were real A friend of mine came across the following on the Web 4 www completedinfinity com Buy a copy of N Contains zero contains the successor of everything it contains contains only these Just 100 Do the math What is the price per

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